A quantum system co-simulation automatic verification method based on partition refinement

Abstract

The application discloses a quantum system mutual simulation automatic verification method based on partition refinement. In view of the difficulty that a traditional method needs to verify infinite quantum states, the application uses the effect semantics in the Heisenberg style, abstracts observable behaviors of a quantum system into effects, and thus gets rid of the dependence on specific input quantum states. The method first converts two quantum systems to be compared into effect labeled transition systems (eLTS) by using a quantum process algebra, wherein states are pairs of programs and cumulative superoperators. Then, a partition refinement algorithm is applied on the eLTS, maximum behavior equivalence classes are iteratively calculated, and a minimized system is constructed. Finally, whether the two minimized systems are isomorphic is judged, so that whether the two quantum programs are mutually simulated can be automatically verified. The application establishes a complete process that can be automatically verified, and is suitable for correctness, security and equivalence verification of quantum communication protocols and concurrent systems.

Description

Technical Field

[0001] This invention belongs to the field of formal verification of quantum programs, specifically relating to an automatic verification method for the behavioral equivalence of quantum programs. More particularly, this invention relates to a partitioning refinement-based technique for verifying whether two quantum programs exhibit indistinguishable behavior under arbitrary quantum inputs, i.e., whether they satisfy the Larsen-Skou mutual analogy relation. Background Technology

[0002] Quantum mutual simulation is an extension of the classical mutual simulation concept in the field of quantum computing, used to verify the behavioral equivalence of different quantum systems. Specifically, two systems are considered mutual simulations if they behave identically in external observation under the same environmental and input conditions. This concept has significant application value in verifying the security of quantum communication protocols. In existing technologies, quantum program calculus often uses probabilistic tag-transfer systems (pLTS) as its semantic model. Under this model, the system state is defined as a configuration... That is, a combination of a specific quantum state and a program term. Two programs P and Q are determined to be mutually analogous if and only if for all possible input quantum states... Configuration and In pLTS, probabilistic mutual analogy relations are satisfied. However, this method has a fundamental flaw: since the input quantum states constitute a continuous and infinite space, verification by enumerating all input states is algorithmically impossible, severely limiting its practical application. To overcome this flaw, recent research has proposed Effect Label Transfer System (eLTS) as an alternative semantic model. eLTS introduces the abstract concept of "effect" to characterize the observable properties of the quantum system and correlates the effect with the observation probability through Born's rule, thus constructing a symbolic semantic framework independent of specific quantum inputs. When all registers are instantiated, eLTS naturally degenerates into pLTS, ensuring consistency with classical probabilistic semantics.

[0003] In terms of behavioral equivalence, Larsen-Skou mutual simulation is a relation based on equivalence classes. It requires that if two states are Larsen-Skou mutual simulations, then for any action, each step of their transition, although potentially leading to different subsequent state distributions, must have the same overall effect in each equivalence class. This definition has good applicability and robustness in the quantum case. Extending the "action-by-action label, equivalence-class-by-equivalence probability aggregation criterion" of classical Larsen-Skou mutual simulation to the effect-labeled transition system (eLTS) allows "probability weights" to be naturally replaced by "effects," thus avoiding dependence on specific quantum input states.

[0004] Furthermore, although the partitioning refinement algorithm within the coalactic framework provides a mathematical foundation for unifying the mutual simulation verification of various systems, when applying this framework and existing algorithms to the quantum system eLTS, the mutual simulation relationship between programs cannot be simply inferred by comparing whether the initial states of two systems are equivalent. Therefore, there is an urgent need in this field for a novel quantum program mutual simulation verification method that can break free from dependence on specific quantum inputs, possess good versatility and high computational efficiency, and correctly handle the unique semantic structure and behavioral characteristics of quantum systems, in order to meet the growing demand for formal verification in the development of quantum communication systems and quantum algorithms. Summary of the Invention

[0005] The purpose of this invention is to propose an automated verification method for mutual simulation of quantum systems based on partition refinement. This method comprises four key parts: first, modeling the quantum communication protocol to be verified and its formal specification using quantum process algebra; second, constructing a symbolic model independent of specific quantum inputs using Heisenberg-style stateless operation semantics, representing the quantum program as an effect label transfer system (eLTS); third, proposing an algorithm based on the naive partition refinement idea, iteratively calculating equivalence class partitions and minimizing the system; and fourth, automatically verifying whether two quantum programs satisfy the Larsen-Skou mutual simulation relation. This method avoids verifying an infinite number of input quantum states, achieving efficient and automated formal verification.

[0006] The specific technical solution for achieving the objective of this invention is as follows:

[0007] An automated verification method for mutual simulation of quantum systems based on partition refinement is characterized by its ability to automatically determine the behavioral equivalence of two quantum systems within a finite number of steps. The method includes the following steps:

[0008] Step 1: Model the target quantum communication protocol using quantum process algebra to obtain a first quantum program and a second quantum program. The first quantum program is the specific program of the quantum communication protocol to be verified, and the second quantum program is the specification program generated according to the formal specification of the protocol.

[0009] Step 2: Based on Heisenberg-style stateless operation semantics, the first quantum program and the second quantum program are respectively transformed into corresponding effect tag transfer systems (eLTS), and the state of the system is denoted as... , of which The system cumulatively applies superoperators, where P is the program term; the transition weights are the effect distributions with a finite support set, and are obtained through tensor product operations with sorting. Maintain consistency between the quantum register dimension and the qubit index order; at the same time, for the constructed eLTS, the partial evaluation rules support the instantiation of some registers of the system into a given partial quantum state;

[0010] Step 3: Apply the partition refinement algorithm to both eLTS instances, iteratively calculating the maximum behavioral equivalence class partition in their respective state spaces. During iteration, the criterion for determining equivalence between two states is: comparing whether the aggregate quantum effects of the successor states falling into the same equivalence class in the current partition are equal under the same action label. The iteration terminates when the equivalence class partition converges. The algorithm returns:

[0011] Maximum behavior equivalence class partitioning: i.e., the final state partitioning after iterative convergence;

[0012] Minimize the system: directly constructed based on the maximum behavior equivalence class partitioning, where each equivalence class becomes a state of the new system;

[0013] Applying the algorithm to the eLTS corresponding to the first quantum program yields the first maximum behavior equivalence class partition and the first minimization system; applying the algorithm to the eLTS corresponding to the second quantum program yields the second maximum behavior equivalence class partition and the second minimization system.

[0014] Step 4: Compare whether the first minimized system and the second minimized system are isomorphic; if they are isomorphic, then determine that the first quantum system and the second quantum system satisfy the Larsen-Skou mutual analogy relation; if they are not isomorphic, then determine that they satisfy the mutual analogy relation.

[0015] Furthermore, the Heisenberg-style stateless semantics described in step 2 specifically include: HPre, HU, HRes, HSum, HPar, and HSync, which map the observable behavior of the quantum program P to an effect-labeled transfer system eLTS(P)=(S,Act,→); where the state set S is the pairing of the superoperator and the program. , The supercomputing applied to the system is accumulated; P is the program item, which records the program segment that has not been executed in the current state; Act represents the set of observable action tags, including silence, input or output classical information, and input or output qubits. This represents a transfer relationship with action labels, where each transfer also carries an effect distribution. The effect distribution associates each successor state s with a quantum effect matrix E; meanwhile, for the constructed eLTS, the partial evaluation rules support the instantiation of a portion of the system's registers into a given partial quantum state. ,in The specific steps include:

[0016] Step 01: Keep the migration structure of the effect label transfer system unchanged;

[0017] Step 02: Adjust the register With density operator Instantiate and distribute the effects carried by the original migration. Updated to ,in For the register to be instantiated, For a specific given quantum state, Represented as register Find the partial trace; For the unit operator of the remaining registers, Operations maintain register index order; when all registers are instantiated, the probabilistic label transfer system pLTS is obtained; when only some registers are instantiated, it remains eLTS.

[0018] Furthermore, the partitioning refinement algorithm described in step 3 takes eLTS as input and outputs the maximum behavior equivalence class partition. The specific steps are as follows:

[0019] Step 11. Initialization step: Divide all states S of the effect labeling transfer system into the same equivalence class. To form the initial partition ;

[0020] Step 12. Iterative Refinement Step: Based on the current partition For each state s in the effect labeling transition system, a standardization operation is performed, and the partition is reconstructed accordingly. This process is iterated until the partition no longer changes, thus obtaining the maximum behavioral equivalence class. The standardized operations include:

[0021] State replacement: Replace each successor state in the successor structure of state s. , replace with In the current division The equivalence class identifier of the following ;

[0022] Effect merging: Extract all action labels from state s. Furthermore, after state replacement, the transitions pointing to the same equivalence class identifier are summed in a matrix to obtain the state s with respect to the equivalence class identifier. The aggregation effect; to all The corresponding aggregation effects are arranged in a uniform order, forming a new successor structure of state s;

[0023] The principle of the reconstruction partition is: if and only if two states are in all When the aggregation effects on all states are completely equal, i.e., when the successor structures are completely identical, these two states are assigned to the same new equivalence class. After traversing all states, a new partition is obtained. .

[0024] Step 13. Termination Determination and Output: Compare the new partition. Compared with the previous round of division If the two are completely identical, the algorithm terminates, and the current partition is the maximum behavior equivalence class partition; otherwise, let Then return to step 12 to continue the next iteration.

[0025] Furthermore, the construction steps of the minimized system described in step 3 are as follows:

[0026] First, the maximum behavior equivalence class is divided. Each equivalence class in the map is mapped to a new state. The set of states that constitutes the minimized system , where k is the number of equivalence classes; for any two states and each action label Perform the following operations:

[0027] Take all From the state of being Transferred to The transition between states is denoted by summing the effect weights as follows: ;like If the value is not zero, then create a migration in the minimal system. .

[0028] Repeat the above process until all state pairs and action labels have been processed, thus constructing the minimized system.

[0029] Furthermore, the isomorphism mentioned in step 4 refers to the isomorphism between action labels and weight effects: there exists a bijection between two sets of minimized system states such that for any label For any objective equivalence class, the effect matrices of the two are equal; under finite states and finite distinguishable effect sets, the two minimization systems are isomorphic if and only if the first quantum program and the second quantum program are equivalent in the sense of Larsen-Skou mutual simulation.

[0030] This invention provides an automated verification method for quantum program mutual simulation based on partitioning refinement, which efficiently determines the behavioral equivalence of quantum programs and overcomes the shortcomings of traditional methods that cannot be algorithmically implemented due to the need to verify an infinite number of quantum input states. The symbolic eLTS model constructed in this invention is independent of specific quantum inputs, and the partitioning refinement algorithm has a time complexity of O(m·n), enabling verification of most small and medium-sized quantum communication protocols and concurrent systems to be completed in seconds. This invention has been successfully applied to the formal verification of the following quantum systems: (1) proving that quantum teleportation protocols and direct transmission of qubits are completely equivalent in behavior; (2) verifying the security of the BB84 quantum key distribution protocol, confirming that the protocol maintains information theory security even in the presence of an eavesdropper. This invention provides an efficient and feasible technical path for the formal verification of quantum software, significantly improving the efficiency and reliability of quantum system design and verification. Attached Figure Description

[0031] Figure 1 This is a flowchart of the present invention;

[0032] Figure 2 This is a pseudocode diagram of constructing eLTS in this invention;

[0033] Figure 3 The pseudocode diagram shows the algorithm for handling concurrent recursive expansion of quantum programs according to the present invention.

[0034] Figure 4 This is a pseudocode diagram of the algorithm for minimizing eLTS based on partition refinement in this invention;

[0035] Figure 5 This is a schematic diagram illustrating the program implementation of the quantum teleportation protocol of the present invention;

[0036] Figure 6 This is a schematic diagram of the eLTS (Elastic Transmission Protocol) derived from Heisenberg's stateless operation semantics in this invention.

[0037] Figure 7 This is a schematic diagram of the partitioning refinement and minimization results of the eLTS stealth protocol according to the present invention;

[0038] Figure 8 This is a schematic diagram of the program implementation of the present invention in the standard procedure (direct transmission of qubits);

[0039] Figure 9 This is a schematic diagram of the canonical program eLTS derived from Heisenberg's stateless operation semantics in this invention;

[0040] Figure 10 This is a schematic diagram illustrating the result of minimizing the partitioning of the standard procedure eLTS in this invention. Detailed Implementation

[0041] The implementation process of the present invention will now be described in detail with reference to specific embodiments and accompanying drawings. Unless otherwise specified, the formal verification theory, quantum semantic model, and partitioning algorithm involved in this invention are all well-known technologies in the field, and the present invention does not impose any additional limitations on them.

[0042] This invention provides an automated verification method for quantum program mutual simulation based on partitioning refinement, the overall process of which is as follows: Figure 1 As shown, the specific steps include:

[0043] Step 1: Model the target quantum communication protocol using quantum process algebra to obtain a first quantum program and a second quantum program. The first quantum program is the specific program of the quantum communication protocol to be verified, and the second quantum program is the canonical program generated according to the formal specification of the protocol. The syntax of the quantum process algebra includes: process termination, nondeterministic selection, conditional selection, concurrency, constraint, measurement, input action, output action, silent action, and sequential execution.

[0044] Step 2: Based on Heisenberg-style stateless operation semantics, the first quantum program and the second quantum program are respectively transformed into corresponding effect tag transfer systems (eLTS), and the state of the system is denoted as... ,in The superoperator applied cumulatively to the system, P being the program term; the transition weights are the effect distributions with a finite support set, and are obtained through tensor product operations with sorting. Maintaining consistency between the quantum register dimension and the qubit index order; simultaneously, for the constructed eLTS, partial evaluation rules support instantiation of some system registers into given partial quantum states; the Heisenberg-style stateless operation semantics are as follows:

[0045] HMeas: The process operates silently, performing destructive measurements and updating configurations and effect distributions;

[0046] HAct: Executes the action tag content and updates the configuration;

[0047] HU: Superoperators accumulated in the current state by left synthesis of superoperators, preserving the effect distribution;

[0048] HRes: Impose restrictions on action labels to achieve scope control while preserving effect distribution;

[0049] HSum: Used to perform nondeterministic choices;

[0050] HCond: Used to execute conditional selection processes; when a condition is met, subsequent procedures are executed.

[0051] HPar: Parallel single-sided step, which merges program items with the structure on the other side, and keeps the superoperator consistent;

[0052] HSync: Aligns synchronous action labels, reduces explicit action labels to silent actions, and forms a joint effect distribution;

[0053] Step 3: Apply the partition refinement algorithm to both eLTS instances, iteratively calculating the maximum behavioral equivalence class partition in their respective state spaces. The criterion for determining equivalence between two states during iteration is: comparing the aggregated quantum effects of their successor states falling into the same equivalence class in the current partition under the same action label. The iteration terminates when the equivalence class partition converges. The algorithm returns the following results:

[0054] Maximum behavior equivalence class partitioning: i.e., the final state partitioning after iterative convergence;

[0055] Minimize the system: directly constructed based on the maximum behavior equivalence class partitioning, where each equivalence class becomes a state of the new system;

[0056] Applying the algorithm to the eLTS corresponding to the first quantum program yields the first maximum behavior equivalence class partition and the first minimization system; applying the algorithm to the eLTS corresponding to the second quantum program yields the second maximum behavior equivalence class partition and the second minimization system.

[0057] Step 4: Compare whether the first minimized system and the second minimized system are isomorphic; if they are isomorphic, then determine that the first quantum system and the second quantum system satisfy the Larsen-Skou mutual analogy relation; if they are not isomorphic, then determine that they satisfy the mutual analogy relation.

[0058] Quantum program eLTS construction algorithms based on effect semantics, such as Figure 2 As shown.

[0059] First, read the complete program text from the source file and divide it into two parts: the file header hdr and the body. At the same time, extract the string string1, which may contain register constraints.

[0060] Parse the initial register set corresponding to each program P in the file header to obtain the mapping relationship InitInfo(P); perform a union operation on the registers of all programs to obtain the global register set Qubit and its number of bits n.

[0061] Next, the operator definition block is extracted from the main part and its syntax is parsed to construct an abstract syntax tree InfoAST containing information such as Kraus operators and superoperators. Based on this, an operator lookup table OpDef is built, which is used to map syntax symbols to specific operator matrices in the future.

[0062] The main body is divided into several program fragment sets Progs according to the def keyword. For each fragment F, it is first split into a program name P and a program body text ctx. Then, the lexer is called to perform lexical analysis on ctx to obtain the token sequence TokList. Based on this, BuildInstrSeq is called to perform a linear scan and reduction on the token sequence, translating the syntax structure into an intermediate instruction sequence Instr[P]. Each instruction corresponds to a silent action, measurement, superoperator, classical input / output, nondeterministic choice, conditional choice, parallel combination, and termination process construction.

[0063] Construct an identity superoperator on the global bit space and initialize the state set and transition set.

[0064] The complete eLTS is generated by concurrently executing the ExploreParallel algorithm for recursive expansion of the quantum program. Finally, the effects of each transition in the eLTS are subjected to a bias trace, thereby obtaining the reduced effect distribution.

[0065] The ExploreParallel algorithm, a concurrent quantum program recursive expansion algorithm, is quite complex and will be described separately here. Figure 3 As shown. This algorithm is used to recursively construct the effect label transfer system (eLTS) of a given concurrent quantum program semantics. Its core objective is to interpret quantum instructions, classical synchronization instructions, and concurrent control structures one by one, based on the current instruction position of each program component, to generate all possible state-effect-action transitions, thereby covering all semantic branches of program execution. Given the concurrent position vector pos, the current state... The algorithm's execution steps are as follows: current program component P, component index p, synchronization queues ShIn and ShOut, trajectory records Res, coefficient table Coeff, measurement table Meas, concurrent program list ParList, and exit set Exit.

[0066] First, read the next instruction for the current component based on the position vector. .

[0067] like If it is a silent action, then the supercomputor in configuration s remains unchanged, and the process... Execution action Silent action It then becomes P, generating migration. Where E is the effect weight corresponding to state s; Self-incrementing And recursively call ExploreParallel with the updated configuration.

[0068] like If it is a measurement command, then different measurement branches are generated according to different measurement operators; the superoperator in the current configuration s is retrieved. Then for each Kraus term of the measurement operator calculate and generate migration , in which configuration The supercomputer was updated to Then update the process; subsequently, continue recursively calling ExploreParallel for each measurement branch.

[0069] like It is a supercomputer Configuration from Updated to .

[0070] like The action is input on channel c, if channel It has appeared In the middle, then read its corresponding value. ,implement If the component is blocked, the recursive expansion continues; otherwise, the scheduling is moved to the next component in the concurrency list. .

[0071] like If the action is output on channel c, then it will... Write If a matching input channel exists, i.e. If the condition is met, then a synchronization operation is performed; subsequently, the process is recursively expanded and execution continues.

[0072] like Therefore The conditions for judgment are selected when... If the recursive expansion continues, then the guard structure is skipped and its internal semantics are not expanded.

[0073] like If the choice is uncertain, then ExploreParallel is called separately for each branch to generate all reachable semantic branches.

[0074] like It terminates the process if all concurrent components satisfy the condition. If the condition is met, the program execution terminates; otherwise, it switches to the next component. .

[0075] Once all reachable paths have been expanded, the algorithm returns the constructed set of states and the set of transitions. .

[0076] eLTS minimization algorithms based on effect semantics, such as Figure 4 As shown. Given an effect label transfer system. The goal is to construct the maximum behavioral equivalence class partition under effect semantics. And generate the corresponding minimal system. The algorithm flow is as follows.

[0077] Step 01. Initialization Phase. First, treat all states as a whole as an equivalence class, denoted as the initial partition. This division was then refined repeatedly until it no longer changed.

[0078] Step 02. Successor structure standardization. For each state... Collect all of its migrations: Each target state Replace each with its current equivalence class number. .

[0079] Then based on the action Equivalence Class Group these successors; for items in the same group, sum their effect matrices: Finally, we obtain the standardized form of this state under the current partition:

[0080] Step 03. Refine the partition. Using a normalized form, repartition the state space according to the following rules: That is, if two states have the same standardized successor structure, they are assigned to the same new equivalence class. Then let... And continue the refinement process until the division is stable.

[0081] Step 04. Minimize system construction. After the partitioning is stable, based on each equivalence class... Introducing a new state For any two equivalence classes and actions The effect between their equivalence classes is obtained by summing the effect matrices of all cross-class transfers: Based on this, a minimum system is defined. Migration relationships:

[0082] Step 05. Output the results. The algorithm ultimately returns the minimized system. and maximum behavior equivalence class partitioning .

[0083] Ultimately, whether the two programs can be mutually simulated is determined based on whether the two minimized systems are isomorphic.

[0084] Taking the quantum teleportation protocol as an example, the implementation effect of this invention is given. Figure 5 This is the source code for a quantum teleportation protocol. In this protocol, Alice has an unknown qubit q1 that she wants to transfer to Bob. Alice and Bob share an entangled pair of qubits q2 (held by Alice) and q3 (held by Bob), initially in the Bell state. Alice performs superoperator operations (CNOT and Hadamard) on q1 and q2, and then measures them. She sends the measurement results (two classical bits) to Bob. Bob performs the corresponding unitary operation (X, Z, or both) on q3 based on the received classical information, thereby reconstructing the state of q1 on q3. The behavior of the entire protocol Tel should be equivalent to: directly passing q1 to Bob and executing B′ on q1, i.e., the canonical procedure of the stealth transfer protocol is as follows: Figure 8 As shown.

[0085] The verification process is as follows: Figure 5 The program shown, after passing through Algorithm 1, yields... Figure 6 As shown in the eLTS diagram, each circle represents a state. Figure 6 The minimized system built after performing the partition refinement algorithm on the eLTS is as follows: Figure 7 As shown, each circle represents an equivalence class, with the state having the smallest index within that class as its representative. Similarly... Figure 8 The program shown, after passing through Algorithm 1, yields... Figure 9 The eLTS shown is in Figure 9 Perform the same partitioning refinement algorithm on the eLTS, construct its minimization system, and obtain Figure 10 .because Figure 7 and Figure 10 It is isomorphic, which shows that the stealth transfer protocol and its standard procedure are mutually simulated.

Claims

1. An automated verification method for mutual simulation of quantum systems based on partitioning refinement, characterized in that, The method includes the following steps: Step 1: Model the target quantum communication protocol using quantum process algebra to obtain a first quantum program and a second quantum program. The first quantum program is the specific program of the quantum communication protocol to be verified, and the second quantum program is the specification program generated according to the formal specification of the protocol. Step 2: Based on Heisenberg-style stateless operation semantics, the first quantum program and the second quantum program are respectively transformed into corresponding effect tag transfer systems (eLTS), and the state of the system is denoted as... ,in The superoperator applied cumulatively to the system, P being the program term; the transfer weights are the effect distributions with a finite support set, and are obtained through a sorted tensor product. The operation maintains the consistency between the quantum register dimension and the qubit index order; at the same time, for the constructed eLTS, the partial evaluation rules support the instantiation of some registers of the system into a given partial quantum state; Step 3: Apply the partition refinement algorithm to both eLTSs, iteratively calculating the maximum behavioral equivalence class partition in their respective state spaces. During iteration, the criterion for determining equivalence between two states is: comparing whether the aggregate quantum effects of the successor states falling into the same equivalence class in the current partition are equal under the same action label. The iteration terminates when the equivalence class partition converges. For each eLTS, the algorithm returns: Maximum behavior equivalence class partitioning: i.e., the final state partitioning after iterative convergence; Minimize the system: Construct a new system directly based on the maximum behavior equivalence class partitioning, where each equivalence class becomes a state of the new system; Applying the algorithm to the eLTS corresponding to the first quantum program yields the first maximum behavioral equivalence class partition and the first minimization system; applying the algorithm to the eLTS corresponding to the second quantum program yields the second maximum behavioral equivalence class partition and the second minimization system. Step 4: Compare whether the first minimized system and the second minimized system are isomorphic; if they are isomorphic, determine that the first quantum program and the second quantum program satisfy the Larsen-Skou mutual simulation relation; if they are not isomorphic, determine that they satisfy the mutual simulation relation.

2. The automatic verification method for mutual simulation of quantum systems as described in claim 1, characterized in that, Step 2, which involves converting the first quantum program and the second quantum program into corresponding effect label transfer systems (eLTS), specifically includes: mapping the observable behavior of the quantum program P to an effect label transfer system eLTS(P) = (S, Act, →) according to semantic rules, including HPre, HU, HRes, HSum, HPar, and HSync; where the state set S is the pairing of the superoperator and the program. , The supercomputing applied to the system is accumulated; P is the program item, which records the program segment that has not been executed in the current state; Act represents the set of observable action tags, including silence, input or output classical information, and input or output qubits. This represents a transfer relationship with action labels, where each transfer also carries an effect distribution. The effect distribution associates each successor state s with a quantum effect matrix E; meanwhile, for the constructed eLTS, the partial evaluation rules support the instantiation of a portion of the system's registers into a given partial quantum state. ,in The specific steps include: Step 01: Keep the migration structure of the effect label transfer system unchanged; Step 02: Adjust the registers With density operator Instantiate and distribute the effects carried by the original migration. Updated to ,in For the register to be instantiated, For a specific given quantum state, Represented as register Find the partial trace; For the unit operator of the remaining registers, Operations maintain register index order; when all registers are instantiated, the probabilistic label transfer system pLTS is obtained; when only some registers are instantiated, it remains eLTS.

3. The automatic verification method for mutual simulation of quantum systems as described in claim 1, characterized in that, Step 3 describes a partitioning refinement algorithm: The specific steps for outputting the maximum behavior equivalence class partition, using eLTS as input, are as follows: Step 11. Initialization step: All states S of the effect label transfer system are assigned to the same equivalence class. To form the initial partition ; Step 12. Iterative Refinement Step: Based on the current partition For each state s in the effect label transfer system, a standardization operation is performed, and the partition is reconstructed accordingly. This process is iterated until the partition no longer changes, resulting in the maximum behavioral equivalence class partition. The standardized operations include: State replacement: Replace each successor state in the successor structure of state s. , replace with In the current division The equivalence class identifier of the following ; Effect merging: Extract all action labels from state s. And after state replacement, they point to the same equivalence class identifier. The migrations, and the matrix summation of the effects carried by these migrations, yield the state s with respect to... The aggregation effect; to all The corresponding aggregation effects are arranged in a uniform order, forming a new successor structure of state s; The principle of the reconstruction partition is: if and only if two states are in all When the aggregation effects on all states are completely equal, i.e., when the successor structures are completely identical, these two states are assigned to the same new equivalence class. After traversing all states, a new partition is obtained. ; Step 13. Termination Determination and Output: Compare the new partition. Compared with the previous round of division If the two are completely identical, the algorithm terminates, and the current partition is the maximum behavior equivalence class partition; otherwise, let Then return to step 12 to continue the next iteration.

4. The automatic verification method for mutual simulation of quantum systems as described in claim 1, characterized in that, The construction steps of the minimized system in step 3 are as follows: First, the maximum behavior equivalence class is divided. Each equivalence class in Mapped to a new state The set of states that constitutes the minimized system , where k is the number of equivalence classes; for any two states and each action label Perform the following operations: Take all from medium state Transferred to The transition between states is denoted by summing the effect weights as follows: ;like If the value is not zero, then create a migration in the minimal system. ; Repeat the above process until all state pairs and action labels have been processed, thus constructing the minimized system.

5. The automatic verification method for mutual simulation of quantum systems as described in claim 1, characterized in that, The isomorphism mentioned in step 4 refers to the isomorphism between action labels and weight effects: there exists a bijection between two sets of minimized system states such that for any label For any objective equivalence class, the effect matrices of the two are equal; under finite states and finite distinguishable effect sets, the two minimization systems are isomorphic if and only if the first quantum program and the second quantum program are equivalent in the sense of Larsen-Skou mutual simulation.

Images